The Universe is expanding, but why aren't we?

Theoretically, the universe is expanding uniformly over time, dragging the galaxies along with it, right? That's why we can see the galaxies flying away from each other faster than the speed of light. But I was told they are not technically moving through space but with it. Like drawings on the surface of a balloon as it is inflated. So if this is true why would this universal (italicized meaning everything) expansion stop at galaxies? Shouldn't suns, planets, molecules, even atoms etc. etc. be expanding away from each other as well?

First, the description of space dragging galaxies along with it isn't an accurate one. General Relativity is used to model what the universe should do at the largest scales. In order to accomplish this we have to "smooth" the universe out and look at it as a whole. If we take the average density of energy and matter and put it in our model, the end result we get says that the universe is expanding.

However, this completely ignores the fact that the universe is NOT homogenous at the small scales. (Which of course is obvious given that we have empty space surrounding very dense objects like stars and planets and such) Calculating what happens within galaxies and gravitationally bound structures requires a completely different calculation. When we do this we get the result that space is NOT expanding within galaxies!

Putting the two together and looking at the evidence (which is the key part of all of this), we see that as a whole the universe is expanding, but locally things will not recede from each other if they are bound together through gravity.

The short and easy version is that the expansion of the universe only affects things which have very little attraction between each other, such as galaxy superclusters. One supercluster is far enough away from another, on average, that they all recede from each other. Within these superclusters individual clusters may be bound, or may not be bound, depending on the cluster. Individual clusters are pretty much always bound together and the galaxies within them are not receding from each other due to expansion.

Theoretically, the universe is expanding uniformly over time, dragging the galaxies along with it, right?

Not really;The acceleration of expansion varies over time...right now it is slowing down a bit. And the further the galaxyfrom us, the faster the velocity of recession relative to us.

But in general relativty, even that statement requires some clarification:Distance in general relativity and cosmology is a bit different than everyday local distance. The 'metric' or distance function is the core idea in the geometry of the universe. Different geometries---different metrics---arise as solutions to the GR equation. In order to define distance one must choose a metric, the 'distance' will be defined in that metric. The standard model is the 'FLRW' metric. In cosmology, one must also move the metric across some defined
spacetime curvature....because we need an agreed upon reference akin to the straight line you use when making everyday local measures with a ruler.

We use a curve of constant time to freeze expansion and get a snapshot. Using such methodology, distant galaxies may be receding 'faster than light'; in other metrics, there is no such superluminal expansion. In cosmology, different models utilize different metrics and so conclude different distances. In other models, there is no such rapid expansion.

That's why we can see the galaxies flying away from each other faster than the speed of light.

Actually, we can't 'see' that...not observe it directly....but it is what is believed to be happening via our models and calculations. For example, check out 'Type 1a supernovas'....those are the standard candles [brightness references]....like a string of 100 watt bulbs..... from which we can calculate/estimate their distances.

But I was told they are not technically moving through space but with it. Like drawings on the surface of a balloon as it is inflated.

Moving with space as envisioned on the surface of a balloon IS a good analogy.

So if this is true why would this universal (italicized meaning everything) expansion stop at galaxies?

There is no sudden cutoff. The normal {metric} accelerated expansion has no effect at all on gravitationally bound systems because it results from an assumption of an isotropic and homogeneous cosmos; Those assumptions do not apply to a lumpy galaxy...the math no longer applies. The cosmological constant {dark energy} , however, does have a slight effect by acting to weaken gravity and this results in a minute increase in orbits, say in our solar system and galaxy, for example.

Calculating what happens within galaxies and gravitationally bound structures requires a completely different calculation. When we do this we get the result that space is NOT expanding within galaxies!

My understanding is that no such calculation can be done....that we cannot solve the GR equations at a lumpy galactic scale....Do you have a source??

edit: The post by Chronos here suggests, but does not confirm, a piece of what I understand:

The above have given very good answers, but I would like to add a bit.

Essentially, there a few different parts of this issue. First is the normal metric expansion. In general relativity, filling up the universe with a homogenous fluid or distribution of matter will cause the universe to expand. The metric that describes the expanding universe is the FRW metric. One key assumption of FRW is homogeneity - galaxies are evenly distributed. However, inside of galaxies, this requirement isn't filled. To describe, say, the solar system, you need to use a different metric that isn't expanding. So, because galaxies are very dense and inhomogeneous, they are NOT affected by metric expansion.

The next factor is dark energy. Dark energy may be a cosmological constant, but it's usually represented by a negative pressure fluid. These are physically equivalent. The thing about negative pressures is that they accelerate the expansion of the universe. So, dark energy does that. However, it also has a tiny effect inside of galaxies, since it exerts a force everywhere.

In summary - metric expansion has no effect inside of galaxies, so space is NOT expanding inside of them. However, dark energy supplies a slight force on everything, however negligible.

Another somewhat related point is that orbits are slightly expanded. However, this isn't due to the expansion of universe, it's due to the warping of spacetime in general relativity by the Sun.

I think you mean that the Friedman equation can not be solved at lumpy galactic scales. GR is a theory of gravity, so you can still use it inside of galaxies. For example, the solution of the Einstein field equation for a spherically symmetric body, such as the sun, a black hole, etc. is the Schwarzschild solution.

But all cosmological solutions do not apply inside of galaxies, correct.

Cosmology has an idea of the background (the nearly uniform soup of light from ancient matter, when space was filled with hot gas). The Hubble law is about the instantaneous rate of expansion of distances between observers who are at rest wrt CMB (cosmic background radiation)

The two opposite edges of a galaxy are not likely to be both at rest wrt CMB.

Therefore the diameter of the galaxy would not be one of the distances to which the law applies. The galaxy is being held together by its own internal forces, so is not subject to Hubble law expansion.

Try to picture the current rate of distance increase. The distance between two stationary observers increases about 1/139 of one percent per million years.*

Things like our solar system and galaxies are not included in this expansion because held together by their own arrays of forces, but if they were it would be very hard to detect.
1/139 of one percent is a very small fractional increase and a million years is a long time to wait before you could see even that much.

*Or if you find it easier to think about it on a 100 million year timescale (a tenth of a billion years) you could say the instantaneous fractional growth rate is1/139 per 108 years.

Greg, there is actually a mental trick for visualizing or conceptualizing. Not anything much really, but it can help to think in terms NOT of billions of years but in units of time which are a TENTH of that size. If you like call such units of time "dimes".
Geological ages tend to be on the order of a dime in length. The Paleozoic Era was 3 dimes long and is normally divided into 6 Ages each close to half a dime.
From the beginning of the Paleozoic up to the present is between 5 and 6 dimes worth of time.
That covers evolution of life from single-cell up to present as if a celestial parking meter (if you want a concrete image) had been fed six dimes.

With that "dime" timescale in mind, now turn to cosmological distance expansion. Largescale distances, between observers or objects which are at rest with respect to the background of Ancient Light, grow at a fractional rate of 1/139 per dime.
Whatever the distance starts out as, at the end of a dime interval it will be just a tiny bit larger: approximately 1/139 bigger.

the fractional rate itself is declining very very slowly, so it will not always be 1/139 per dime. Way far in future it will get down around 1/163 per dime and become nearly constant (according to the usual cosmo model.) This leveling out at 1/163 instead of continuing to decline is the effect of "acceleration". A distance that grows at a constant fractional rate (year after year or dime after dime ) is of course growing exponentially, along an ever-steepening curve.

I find visualizing in terms of that time unit helps me think about the process. Let me know if it totally confuses you or draws a blank or helps some.

It seems to me as a layman that space must be expanding everywhere, including within galaxies. It's just that the effect of gravity overwhelms the universal expansion so that in calculating the distance between objects the expansion can be ignored. On the other hand, gravity deforms spacetime, so maybe space isn't expanding within a galaxy in any sense of the word.

But let's say that there's a cube of space, with a galaxy in the center of it, and the cube is large enough that the effect of gravity at the cube's edges is minimal. Presumably the edges expand at the usual cosmological rate, right? Do the centers of the edges of the cube get deformed toward the galaxy at the center? If so, that would make the expansion of space pretty lumpy.

One thing that's difficult to understand is exactly what expansion of space means. Here's a question that may help me grasp it. Let's say that we measure the speed of light between two galaxies. Then we wait a million years and measure it again, after space expansion has moved those galaxies further apart. Does it then take longer for light to make that same trip? In other words, is the speed of light constant as measured per today's kilometer - so that it takes longer - or is it constant as measured per spatially lengthened kilometer - so that it takes the same time? Thanks.

One thing that's difficult to understand is exactly what expansion of space means.

Good. GR does not say that space is a thing, or a material, or a fabric. It talks about geometry (distances between observers, events, geometric measurements).

And in cosmology we have a concept of an observer being at rest wrt background---the ancient light, and the expansion process itself, appear the same in all directions to such an observer. That's what being at rest means--no doppler effects of moving in any direction. Universe time is time as measured by observers who are at universe-rest.

Hubble Law expansion simply says that at any given moment of universe time, distances between stationary observers are growing at a given fractional rate. At this time the rate is estimated to be about 1/139 of one percent per million years.

GR is about geometry itself. Distances areas angles volumes and how they are related. In some sense it should not be thought of as substantive or as physical material because geometry in a way is more fundamental than matter. The particles and fields of matter take place in a framework of geometry. Our understanding of material behavior is in a frame of geometry. So material analogies have only limited value.

Here's a question that may help me grasp it. Let's say that we measure the speed of light between two galaxies. Then we wait a million years and measure it again, after space expansion has moved those galaxies further apart. Does it then take longer for light to make that same trip? In other words, is the speed of light constant as measured per today's kilometer - so that it takes longer -...

Yes that's right! After a million years the distance is longer so of course the light takes longer!
Proper distances (used in cosmology and in formulating Hubble Law of expansion) are actually defined as the distance at a particular moment in Universe time. As if you could freeze expansion to allow you to measure without the distance changing while you were measuring.
So after a million years the distance would be 1/139 percent longer and therefore the light would take correspondingly longer to travel the distance. Our measurement units are not affected.

In the larger picture, galaxies tend to be approximately at rest with respect to background. A few hundred km/s random local motion, which can be neglected when we discuss stuff at large scale. So in answering I assume that these galaxies are like the observers I spoke of: at universe rest.

If you don't specify that you are talking about freezeframe (proper) distance then complications can arise about what your example means and how it relates to Hubble Law. Over very long time intervals the fractional expansion rate itself, the 1/139, changes so you can get into a lot of blahblahblah over details. But basically what you said is absolutely right. After a million years the distance is longer so a light pulse would take correspondingly longer.